A381648 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A381649.

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, 44, 0, 1, 4, 18, 98, 510, 0, 1, 5, 26, 163, 1133, 7024, 0, 1, 6, 35, 240, 1884, 15508, 109362, 0, 1, 7, 45, 330, 2779, 25659, 239808, 1871530, 0, 1, 8, 56, 434, 3835, 37704, 394313, 4076904, 34590180, 0, 1, 9, 68, 553, 5070, 51891, 576178, 6661602, 74895252, 682396379, 0

Offset

0,8

Links

Table of n, a(n) for n=0..65.

Formula

A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-j+k,j)/(3*n-j+k) * A(n-j,3*j).

Example

Square array begins:
  1,      1,      1,      1,      1,      1,       1, ...
  0,      1,      2,      3,      4,      5,       6, ...
  0,      5,     11,     18,     26,     35,      45, ...
  0,     44,     98,    163,    240,    330,     434, ...
  0,    510,   1133,   1884,   2779,   3835,    5070, ...
  0,   7024,  15508,  25659,  37704,  51891,   68490, ...
  0, 109362, 239808, 394313, 576178, 789055, 1036973, ...

Prog

(PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-j+k, j)/(3*n-j+k)*a(n-j, 3*j)));

Crossrefs

Columns k=0..1 give A000007, A381649.

Cf. A381594, A381603.

Sequence in context: A118349 A384620 A381592 * A384808 A384811 A385061

Adjacent sequences: A381645 A381646 A381647 * A381649 A381650 A381651

Keyword

nonn,tabl

Author

Seiichi Manyama, Mar 03 2025

Status

approved